(a) Prove the identity

$$\nabla(\mathbf{F} \cdot \mathbf{G})=(\mathbf{F} \cdot \nabla) \mathbf{G}+(\mathbf{G} \cdot \nabla) \mathbf{F}+\mathbf{F} \times(\nabla \times \mathbf{G})+\mathbf{G} \times(\nabla \times \mathbf{F})$$

(b) If $\mathbf{E}$ is an irrotational vector field (i.e. $\nabla \times \mathbf{E}=\mathbf{0}$ everywhere), prove that there exists a scalar potential $\phi(\mathbf{x})$ such that $\mathbf{E}=-\nabla \phi$.

Show that the vector field

$$\left(x y^{2} z e^{-x^{2} z},-y e^{-x^{2} z}, \frac{1}{2} x^{2} y^{2} e^{-x^{2} z}\right)$$

is irrotational, and determine the corresponding potential $\phi$.